The idea of a convolutional double copy goes back to [1]},
where the global U-duality groups are analysed. However, more significantly
it is shown in [2]} that the convolutional product of gauge theory
symmetries reproduces the symmetries of the gravitational theory at the
linear level. That there may be a convolutional square relation between
gauge theory and gravitational fields in position space is not so
surprising if we recall the double copy relation for scattering amplitudes,
which are derived in momentum space. In [3]}, taking the
low-energy limit of the bosonic string (\(\mathcal {N}=0\)  supergravity) as a
concrete gravitational theory, it is found that in order to have a matching
of the degrees of freedom in this theory, i.e. (\(D^2-2D+2\) ), and the
square of the Yang-Mills theory, i.e. \((D-1)^2\) , as well as to be able to
disentangle the dilaton and the trace of the metric, one ought to consider a
BRST covariant formulation
for each of the theories. Thus, a set of ghost fields
are introduced for the gravitational theory, as well as the gauge theories.
This sets up a dynamical equivalence between gauge theories squared and
gravitational theories.The idea that BRST ghosts are needed
in order to establish the double copy relations was presaged in the
work of Siegel in [4]}.
However, an important assumption in this
construction is that all fields are sourced by non-trivial currents, which
makes it possible formally to invert the d'Alembertian operator appearing
in the equations of motion.
